Blowups of triangle-free graphs

TL;DR

The authors resolve the optimal γ-dependence in Nikiforov's blowup theorem for the important class of triangle-free graphs by showing that any $n$-vertex graph containing $ ilde{c}_H(eta)n^h$ copies of a triangle-free $H$ must contain a blowup $H[k]$ with $k$ on the order of $rac{ ext{log} n}{ ext{log}(1/eta)}$. They introduce a density-increment method combined with a product-structure property guaranteed by triangle-freeness and a dependent random choice technique to inductively build the blowup. As a corollary, they obtain multicolor Ramsey bounds for blowups of triangle-free graphs: for large enough blowups, the Ramsey number $r(H[k];q)$ grows polynomially in $q$, aligning blowups of triangle-free graphs with bipartite behavior in Ramsey theory. These results yield new power-saving bounds for extremal problems of non-bipartite graphs and open up progress toward extending the optimal γ-dependence to broader classes of graphs. Overall, the paper advances our understanding of how local copies of triangle-free graphs enforce large structured subgraphs and clarifies the Ramsey-theoretic implications of blowups in this regime.

Abstract

A highly influential result of Nikiforov states that if an $n$-vertex graph $G$ contains at least $γn^h$ copies of a fixed $h$-vertex graph $H$, then $G$ contains a blowup of $H$ of order $Ω_{γ,H}(\log n)$. While the dependence on $n$ is optimal, the correct dependence on $γ$ is unknown; all known proofs yield bounds that are polynomial in $γ$, but the best known upper bound, coming from random graphs, is only logarithmic in $γ$. It is a major open problem to narrow this gap. We prove that if $H$ is triangle-free, then the logarithmic behavior of the upper bound is the truth. That is, under the assumptions above, $G$ contains a blowup of $H$ of order $Ω_H (\log n/{\log(1/γ)})$. This is the first non-trivial instance where the optimal dependence in Nikiforov's theorem is known. As a consequence, we also prove an upper bound on multicolor Ramsey numbers of blowups of triangle-free graphs, proving that the dependence on the number of colors is polynomial once the blowup is sufficiently large. This shows that, from the perspective of multicolor Ramsey numbers, blowups of fixed triangle-free graphs behave like bipartite graphs.

Theorems & Definitions (31)

• Theorem 1.1: Nikiforov MR2409174MR2398823
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Conjecture 1.6
• Definition 3.1
• Lemma 3.2
• Theorem 3.3
• proof